Newspace parameters
| Level: | \( N \) | \(=\) | \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 990.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(158.779981615\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 462x + 3072 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 110) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 462x + 3072 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 8\nu - 311 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} - 8\beta _1 + 925 ) / 3 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
4.00000 | 0 | 16.0000 | −25.0000 | 0 | −142.542 | 64.0000 | 0 | −100.000 | |||||||||||||||||||||||||||
| 1.2 | 4.00000 | 0 | 16.0000 | −25.0000 | 0 | 170.240 | 64.0000 | 0 | −100.000 | ||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 0 | 16.0000 | −25.0000 | 0 | 245.302 | 64.0000 | 0 | −100.000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(11\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 990.6.a.ba | 3 | |
| 3.b | odd | 2 | 1 | 110.6.a.g | ✓ | 3 | |
| 12.b | even | 2 | 1 | 880.6.a.k | 3 | ||
| 15.d | odd | 2 | 1 | 550.6.a.l | 3 | ||
| 15.e | even | 4 | 2 | 550.6.b.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.6.a.g | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 550.6.a.l | 3 | 15.d | odd | 2 | 1 | ||
| 550.6.b.k | 6 | 15.e | even | 4 | 2 | ||
| 880.6.a.k | 3 | 12.b | even | 2 | 1 | ||
| 990.6.a.ba | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(990))\):
|
\( T_{7}^{3} - 273T_{7}^{2} - 17472T_{7} + 5952592 \)
|
|
\( T_{17}^{3} - 627T_{17}^{2} - 2850906T_{17} + 1839018960 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{3} \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( (T + 25)^{3} \)
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| $7$ |
\( T^{3} - 273 T^{2} + \cdots + 5952592 \)
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| $11$ |
\( (T - 121)^{3} \)
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| $13$ |
\( T^{3} + 204 T^{2} + \cdots - 340553840 \)
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| $17$ |
\( T^{3} + \cdots + 1839018960 \)
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| $19$ |
\( T^{3} + \cdots - 5618777552 \)
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| $23$ |
\( T^{3} + \cdots - 3774572208 \)
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| $29$ |
\( T^{3} + \cdots - 18666595548 \)
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| $31$ |
\( T^{3} + \cdots - 170764182080 \)
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| $37$ |
\( T^{3} + \cdots + 566059440364 \)
|
| $41$ |
\( T^{3} + \cdots - 25758181944 \)
|
| $43$ |
\( T^{3} + \cdots + 5284163292160 \)
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| $47$ |
\( T^{3} + \cdots - 20675131920 \)
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| $53$ |
\( T^{3} + \cdots - 12560852819916 \)
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| $59$ |
\( T^{3} + \cdots + 282883115232 \)
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| $61$ |
\( T^{3} + \cdots + 48702632528260 \)
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| $67$ |
\( T^{3} + \cdots - 6885873035264 \)
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| $71$ |
\( T^{3} + \cdots - 16794837141696 \)
|
| $73$ |
\( T^{3} + \cdots + 63435730635232 \)
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| $79$ |
\( T^{3} + \cdots + 455163121410304 \)
|
| $83$ |
\( T^{3} + \cdots + 37489435741440 \)
|
| $89$ |
\( T^{3} + \cdots + 64106131539132 \)
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| $97$ |
\( T^{3} + \cdots + 43490447216512 \)
|
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